Add tests for second derivatives

test/finitedifferences.jl

@@ -21,6 +21,9 @@ using FiniteDifferences
         @testset "Jacobian" begin
             test_jacobians(backend)
         end
+        @testset "Second derivative" begin
+            test_second_derivatives(backend)
+        end
         @testset "Hessian" begin
             test_hessians(backend)
         end

test/forwarddiff.jl

@@ -19,6 +19,9 @@ using ForwardDiff
         @testset "Jacobian" begin
             test_jacobians(backend)
         end
+        @testset "Second derivative" begin
+            test_second_derivatives(backend)
+        end
         @testset "Hessian" begin
             test_hessians(backend)
         end

test/reversediff.jl

@@ -14,6 +14,9 @@ using ReverseDiff
         @testset "Jacobian" begin
             test_jacobians(backend)
         end
+        @testset "Second derivative" begin
+            test_second_derivatives(backend)
+        end
         @testset "Hessian" begin
             test_hessians(backend)
         end

test/test_utils.jl

@@ -6,6 +6,7 @@ Random.seed!(1234)
 fder(x, y) = exp(y) * x + y * log(x)
 dfderdx(x, y) = exp(y) + y * 1 / x
 dfderdy(x, y) = exp(y) * x + log(x)
+dfderdxdx(x, y) = -y * 1 / x^2
 
 fgrad(x, y) = prod(x) + sum(y ./ (1:length(y)))
 dfgraddx(x, y) = prod(x) ./ x
@@ -143,6 +144,49 @@ function test_jacobians(backend; multiple_inputs=true, test_types=true)
     @test yvec == yvec2
 end
 
+function test_second_derivatives(backend; multiple_inputs=false, test_types=true)
+    if multiple_inputs
+        # ... but 
+        error("multiple_inputs=true is not supported.")
+    else
+        # explicit test that AbstractDifferentiation throws an error
+        # don't support tuple of second derivatives
+        @test_throws ArgumentError AD.second_derivative(
+            backend, x -> fder(x, yscalar), (xscalar, yscalar)
+        )
+        @test_throws MethodError AD.second_derivative(
+            backend, x -> fder(x, yscalar), xscalar, yscalar
+        )
+    end
+
+    # test if single input (no tuple works)
+    dder1 = AD.second_derivative(backend, x -> fder(x, yscalar), xscalar)
+    if test_types
+        @test dder1[1] isa Float64
+    end
+    @test dfderdxdx(xscalar, yscalar) ≈ dder1[1] atol = 1e-8
+    valscalar, dder2 = AD.value_and_second_derivative(
+        backend, x -> fder(x, yscalar), xscalar
+    )
+    if test_types
+        @test valscalar isa Float64
+        @test dder2[1] isa Float64
+    end
+    @test valscalar == fder(xscalar, yscalar)
+    @test norm.(dder2 .- dder1) == (0,)
+    valscalar, der, dder3 = AD.value_and_derivatives(
+        backend, x -> fder(x, yscalar), xscalar
+    )
+    if test_types
+        @test valscalar isa Float64
+        @test der[1] isa Float64
+        @test dder3[1] isa Float64
+    end
+    @test valscalar == fder(xscalar, yscalar)
+    @test norm.(der .- AD.derivative(backend, x -> fder(x, yscalar), xscalar)) == (0,)
+    @test norm.(dder3 .- dder1) == (0,)
+end
+
 function test_hessians(backend; multiple_inputs=false, test_types=true)
     if multiple_inputs
         # ... but